J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
Nonlinear shear and extensional flow dynamics
of wormlike surfactant solutions
B. Yesilata a,b,∗ , C. Clasen a,c , G.H. McKinley a
a
Department of Mechanical Engineering, Hatsopoulos Microfluid Laboratory, M.I.T., Cambridge, MA 02139, USA
b Department of Mechanical Engineering, Harran University, Osmanbey Campus, Sanliurfa 63300, Turkey
c Institute of Technical and Macromolecular Chemistry, University of Hamburg, 22041 Hamburg, Germany
Received 5 June 2005; received in revised form 4 October 2005; accepted 7 October 2005
Abstract
Nonlinear shear and extensional flow dynamics of a wormlike micellar solution based on erucyl bis(2-hydroxyethyl) methyl ammonium chloride
(EHAC) are reported here. The influences of surfactant (EHAC) and salt (NH4 Cl) concentrations on the linear viscoelastic parameters are determined
using small amplitude oscillatory shear experiments. The steady and time-dependent shear rheology is determined in a double gap Couette cell,
and transient extensional flow measurements are performed in a capillary breakup extensional rheometer (CABER). In the nonlinear shear flow
experiments, the micellar fluid samples show strong hysteretic behavior upon increasing and decreasing the imposed shear stress due to the
development of shear-banding instabilities. The non-monotone flow curves of stress versus shear rate can be successfully modeled in a macroscopic
sense by using the single-mode Giesekus constitutive equation. The temporal evolution of the flow structure of the surfactant solutions in the Couette
flow geometry is analyzed by instantaneous shear rate measurements for various values of controlled shear stress, along with FFT analysis. The
results indicate that the steady flow bifurcates to a global time-dependent state as soon as the shear-banding/hysteresis regime is reached. Increasing
the salt–surfactant ratio or the temperature is found to stabilize the flow, and corresponds to decreasing values of anisotropy factor in the Giesekus
model. Finally, we have investigated the dynamics of capillary breakup of the micellar fluid samples in uniaxial extensional flow. The filament
thinning behavior of the micellar fluid samples is also accurately predicted by the Giesekus constitutive equation. Indeed quantitative agreement
between the experimental and numerical results can be obtained providing that the relaxation time of the wormlike micellar solutions in extensional
flows is a factor of three lower than in shear flows.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Nonlinear; Wormlike micelles; Extensional flow; Capillary breakup extensional rheometry; Shear banding; Hysteresis
1. Introduction
Surfactant molecules consist of a hydrophilic head group and
a hydrophobic tail and under certain conditions they can spontaneously self-assemble into long, flexible wormlike micelles in
aqueous solutions. The structure and rheology of these micelles
are extremely sensitive to surfactant and counterion concentration as well as to temperature because the individual micelles
are constantly being destroyed and recreated through Brownian
fluctuations. Wormlike micellar systems are therefore widely
used in industry as viscosity modifiers and enhancers [1,2]. They
also offer potential as drag-reduction additives in district heating
systems [3]. Recently, wormlike micellar systems have been suc-
∗
Corresponding author. Tel.: +90 536 7360900; fax: +90 414 3440031.
E-mail address: [email protected] (B. Yesilata).
0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jnnfm.2005.10.009
cessfully used in separation of DNA fragments [4] and in forming templates for nanostructures [5]. They are also used as rheology and flow control agents in petroleum transport systems [6].
The main advantage of these liquids for enhanced oil recovery
is their ability to undergo dramatic structural changes upon contacting hydrocarbons, leading to an important drop in “viscosity”
(after completing its fracturing task, this liquid can be easily
removed from the parts of the fracture contacting with hydrocarbon). The fracture cleanup is therefore greatly improved
especially in production zones. The ability to easily formulate
the fluid outdoors makes these liquids very attractive and commercially successful for oil extraction applications [7–10].
A cationic surfactant, erucyl bis(2-hydroxyethyl) methyl
amomonium chloride (EHAC), has very recently been used in
several studies with different counterions (EHAC/NaSal/Water,
EHAC/NaCl/Water and EHAC/NaTos/Water [11–13]; EHAC/2propanol/KCl/Water [14] and EHAC/KCl/Water [15]). Ragha-
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B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
van and Kaler [11] have reported that EHAC-based solutions
can form an entangled network of extremely long wormlike
micellar chains and become highly viscoelastic or gel-like with
increasing salt concentration. The ability of these surfactant
systems to form long, polymerlike structures, but with the additional possibility of breakage and reformation at much lower
scission energy, leads to the terminology of “living” or “equilibrium” polymers (for recent reviews on the rheology of wormlike
micelles see Refs. [16–18]). EHAC solutions show robust viscoelastic behavior even at high temperatures [11,12] in comparison to other viscoelastic surfactant systems, which is important
in oilfield applications.
The linear viscoelastic rheological properties of these viscoelastic wormlike micelles in brine solutions can usually be
characterized, at least at low frequencies, with a single relaxation time λ and they are therefore well described by a simple
Maxwell model. This narrowing of the viscoelastic spectrum is
observed when the micellar breaking time λbr is short compared
to the diffusion or reptation timescale λrep of the whole micelle
[19]. Deviations from this monoexponential relaxation behavior
are observed at high frequencies due to Rouse-like behavior of
the micellar segments [20–22].
The nonlinear steady shear rheology of wormlike micelles
shows a much more complex behavior as indicated schematically in Fig. 1. The dimensionless shear stress (τ * = τ xy /G0 ) is
plotted as a function of dimensionless shear rate (γ ∗ = γ̇xy λ).
The zero shear rate viscosity is given by η0 = G0 λ. The stress first
rises with a slope of unity and then deviates at a shear rate of γ1∗ .
The stress falls with increasing shear rate after passing through a
∗ at a shear rate γ ∗ . This maximum has been
maximum stress τmax
M
attributed by Cates and co-workers (within the scope of reptation
theory) to be the maximum stress that a reptating micellar tube
segment can sustain [19,23,24]. An increasing destruction rate
of the tube segments due to retraction of the wormlike micelles
with further increases in the shear rate results in a falling stress
∗ . However, the decreasing shear stress
at shear rates γ ∗ > γM
region cannot persist to infinite shear rates, and eventually an
Fig. 1. A representative nonmonotonic dimensionless stress-deformation rate
relation in shear flow of a wormlike micellar solution.
upturn in the shear stress curve is observed due to an increasing
solvent contribution to the stress in the fluid sample until the
∗ is reached again at a shear rate γ ∗ . The separation of
stress τmax
2
these two linear regions depends on the solvent viscosity ratio
βs = ηs /η0 .
Cates and co-workers [19,23,24] also developed a constitutive equation for micellar solutions based on the reptation model
for steady flows with a stress tensor τ given by:
1
15
τ=
G0 W − I
(1)
4
3
where W is the second moment of the orientational distribution
function [24]. Although this equation is able to predict the maximum behavior of the stress, it fails to capture the subsequent
upturn of the stress from the solvent and disentangled chain segments. However, the nonlinear differential constitutive equation
proposed by Giesekus [25,26], originally developed from a network theory for entangled polymer systems, has proven to give
a very good description of the first stress maximum as well as
the upturn at high shear rates [27,28]. The Giesekus model for
a single relaxation mode can be written as:
λ
I + α τ p · τ p + λτ p(1) = ηp ␥˙
ηp
(2)
␶s = ηs ␥˙
The symbol τ p(1) denotes the upper convected derivative of
the stress tensor and γ̇ the rate of strain tensor and τ is the total
stress given by τ = τ p + τ s . Although the single-mode Giesekus
model is a semi-empirical constitutive equation (incorporating
an adjustable anisotropy factor α), its “spectacularly successful
description of semidilute wormlike micelles” [1] in nonlinear
shear flow makes it a very useful tool in the description of the
reported hysteresis behavior of the flow curves in the present
paper. Recent experiments using another gel-like surfactant system also show excellent agreement with the predictions of the
Giesekus model in transient shear flow [29].
∗ < γ∗ < γ∗ , a
If an applied shear rate lies in the regime γM
m
homogenous flow can no longer be stable and the system evolves
towards a new stationary state, in which the system forms one
or more “shear-bands” [30–32]. Thus, a steady shear flow can
only be supported by separate regions of fluid flowing at the
high γ2∗ and low γ1∗ shear rate limits of the stress plateau region
indicated by the dotted line in Fig. 1. According to this picture,
there is no constraint on the number of band configurations, and
a set of multiple stationary flow states is possible in a shear
rate controlled situation. Still, the stress in this plateau regime
is uniform throughout the shear-bands and at the interface(s)
between bands. In order to select the number and location of
these bands a higher order theory is required, which incorporates
spatial fluctuations across the gap in the number of density or
concentrations of the micellar species [31–33].
An even more complex rheological response is expected for
the case of a shear induced structure (SIS) formation (such as
gelation [34] or liquid crystalline phases [35,36]), which can
occur in a micellar solution. Such dynamical transitions have
been theoretically described by Olmsted and co-workers [33,37]
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
and also monitored experimentally by Fuller and co-workers
[38,39].
In the following report we will focus on a relatively new
surfactant system with a pronounced viscosity enhancement
(several orders of magnitude higher than that for comparable
systems) and with unusually rich equilibrium phase behavior.
The corresponding surfactants exhibit both shear-induced phase
separation (SIPS) and nonzero dichroism signal under shear
[13]. In these systems, a slowly increasing shear rate ramp is
expected to result in a stable flow up to the maximum stress
∗
τmax
in the plateau level as indicated by the dashed line in
Fig. 1 [24,40] and this typical stress behavior of an up-ramp
shear rate profile has been reported by several authors [41,42].
However, beyond the critical shear rate, the stress level corresponding to the plateau region can lie between the maximum
∗ at the local minimum in Fig. 1.
∗
stress τmax
and the stress τmin
Fischer and co-workers [38,43,44] demonstrated that the dynamics of the inhomogeneous shear-band layers coexisting at such
stress levels could also lead to periodic oscillatory fluctuations of
the stress around this average plateau level. Very recent studies
[39,45–48] have also confirmed that many shear-banding systems display periodic oscillations and time-chaotic fluctuations
in their bulk rheology, rheo-optics or velocimetry. It may thus
be anticipated that the final stationary stress state of the flow in
the plateau regime is strongly history-dependent and a hysteresis between the measured macroscopic behavior upon increasing
and decreasing the shear rate is expected [49,50]. However, there
have been no detailed experiments reported of such phenomena.
We will examine this hysteretic behavior under controlled stress
conditions, for EHAC-based wormlike micellar fluid samples.
The extensional flow behavior of viscoelastic wormlike
micelles has received far less attention than studies of the shear
rheology to date. However, recent reports on flow phenomena
occurring in wormlike micellar solutions, such as rising bubbles
in micellar solutions [51], falling spheres and flow past spheres
[52,53] or filament rupture [54,55] of wormlike micellar solutions have taken into account not only the shear flow properties,
but also the steady and transient elongational viscoelastic properties of the solution.
The first investigations of the apparent extensional viscosity
of wormlike micellar solutions were conducted by Prudhomme
and Warr [56], Walker et al. [57], Fischer et al. [58] and Lu et
al. [59], using the opposed jet device. They reported a general
thickening of wormlike micellar solutions with imposed extension rate in contrast to the reported shear thinning behavior.
An observed drop of the extensional viscosity at high extension
rates was explained by Chen and Warr [60] by micellar scission
at high rates and supported with measurements of the radius of
gyration in the extensional flow field of the opposed jet device.
However, severe problems in the quantitative correlation of the
material functions determined with the opposed jet device [61]
have led to other methods for studying extensional flows. Kato et
al. [62] used rheo-optical studies in a four roller mill, and Muller
et al. [63] investigated the flow through porous media to measure steady elongational viscosities. For robust determination of
the transient elongational viscosity, wormlike micellar solutions
were investigated recently with a capillary breakup rheometer
75
by Anderson et al. [64], using the EHAC system reported on
in the current paper, and with a filament stretching rheometer
by Rothstein [65], who was able to calculate a scission energy
for wormlike micelles in strong extensional flows, supporting
the hypothesis of Chen and Warr [60] for the drop in the extensional viscosity at high rates.
In the present study, we examine the hysteretic behavior of
wormlike micellar fluid solutions in nonlinear shear experiments
upon increasing and decreasing the deformation rate under controlled shear stress conditions for different temperatures and salt
concentrations. We compare these results with the theoretical
predictions of the single-mode Giesekus model using parameters
that are independently determined from linear viscoelastic data
obtained using small amplitude oscillatory flow experiments.
The Giesekus equation is then used to predict the behavior of
wormlike micellar solutions in a uniaxial extensional flow field.
These theoretical results are then compared to experimental
measurements of the transient extensional response of wormlike micellar solutions determined using the capillary breakup
extensional rheometer (CABER).
2. Experimental
2.1. Apparatus
The oscillatory and steady shear flow measurements were
performed with an AR 2000 stress controlled rheometer (T.A.
Instruments, Newcastle DE, USA) in a double gap concentric
cylinder fixture (rotor outer radius: 21.96 mm, rotor inner radius:
20.38 mm, gap: 0.5 mm, approximate sample volume: 6.48 ml
and cylinder immersed height: 59.5 mm). Extensional flow measurements were conducted in a capillary breakup extensional
rheometer developed in collaboration with the Cambridge Polymer Group (Cambridge, MA, USA). The device holds a fluid
sample between circular plates (with radius of 3 mm, initial plate
separation of 2.2 mm). An axial step strain is imposed by separating the plates rapidly within 50 ms to a final separation of
6.6 mm. The midpoint diameter of the fluid filament is monitored
using a laser micrometer with a calibrated minimum resolution
of 20 ␮m. The global evolution of the column profile is also
recorded with a standard CCD camera. More detailed description about the CABER device is given elsewhere [66–68].
2.2. Fluid samples
The surfactant solution, a mixture of erucyl bis(2hydroxyethyl) methyl ammonium chloride and iso-propanol
(25 wt%), is obtained from Schlumberger Cambridge Research.
The fluid samples to be studied were diluted with an appropriate amount of brine solution (ammonium chloride in deionized
water) and stirred to homogeneity for 72 h. The samples are then
kept at rest for a time-period of 7 days prior to the measurements.
Two sets of fluid samples were prepared to investigate the
effects of EHAC concentration (Csurf ) and the molar concentration ratio of NH4 Cl/EHAC (C* = Csalt /Csurf ). For the first set
of six samples the EHAC concentration (54 mM or 2.25 wt%)
was held constant while NH4 Cl concentrations were varied from
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B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
143 mM (0.75 wt%) to 858 mM (4.5 wt%) with increments of
143 mM (0.75 wt%). For the second set of five samples, the
molar concentration ratio of NH4 Cl/EHAC was kept constant
(C* = 10.6) and the surfactant concentration was varied from
18 mM (0.75 wt%) to 90 mM (3.75 wt%) with an increment of
18 mM (0.75 wt%).
age lifetime) of the micelles is much smaller than the reptation
time (the diffusion time). Numerous breaking and re-formation
processes occur within the time scale of the overall relaxation
and consequently average out the relaxation process leading to
a pure, monoexponential stress decay. On the basis of reptation
theory [70], Cates [23] developed an expression for this single
relaxation time depending on the breakup and reptation time:
3. Results and discussion
λ = (λbr λrep )0.5 .
3.1. Linear viscoelastic behavior of micellar solutions
The linear viscoelastic (LVE) behavior of complex fluids can
be described in terms of the complex relaxation modulus G*
It is now well known that viscoelastic surfactant solutions can
behave, under certain conditions, like an ideal Maxwell material
with a single characteristic relaxation time, λ, for the whole
system. However, the linear rheological properties of wormlike
micellar solutions depend on the ratio of breaking and reptation
times, denoted
λbr
ζ=
λrep
(3)
where the weak micellar structures break and recombine on an
individual timescale denoted λbr . According to Cates and coworkers [19,20,23,69] a single exponential relaxation is only
observed if λbr λrep . In this case the breaking time (the aver-
G∗ (ω) = G (ω) + iG (ω)
(4)
(5)
where the storage modulus, G (ω), and the loss modulus, G (ω),
are expressed for an ideal Maxwell model with a single relaxation time by
G (ω) = G0
ω 2 λ2
1 + ω 2 λ2
(6)
ωλ
.
1 + ω 2 λ2
(7)
G (ω) = G0
In Fig. 2, we show the remarkable agreement of the experimental data (T = 25 ◦ C) at low and medium frequencies with
Fig. 2. The storage moduli, G (ω), and the loss moduli, G (ω), determined in a frequency range of 0.01–100 rad/s with a constant shear strain of 10% for the EHAC
solutions of: (a) Csurf = 90 mM; C* = 10.6, (b) Csurf = 72 mM; C* = 10.6, (c) Csurf = 54 mM; C* = 10.6 and (d) Csurf = 54 mM; C* = 16 (experiments were performed at
T = 25 ◦ C).
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
this single relaxation time Maxwell model. The deviations from
Maxwellian behavior in general start at higher frequencies for
larger values of Csurf (for constant C* ) and larger values of C*
(for constant Csurf ). The relaxation time λ and the plateau modulus G0 were determined from a least squares fit of the Maxwell
model (Eqs. (6) and (7)) to the measured LVE data. This was
done using a Cole–Cole representation of the storage and loss
moduli as shown in Fig. 3. In these plots, the loss modulus is
plotted as a function of the storage modulus, and this enables a
more precise determination of the relaxation behavior of samples
than simple frequency sweeps [20]. Semi-circular profiles are
obtained from the Maxwell model and represent the pure monoexponential stress relaxation behavior at T = 25 ◦ C. Fig. 3(a)
shows the effect of the surfactant concentration (Csurf ) on the
linear viscoelastic behavior for a constant salt–surfactant concentration ratio of C* = 10.6, whereas Fig. 3(b) shows the effects
of varying salt concentration at a constant surfactant concentration of Csurf = 54 mM. The diameter of the semi-circular shape
is a measure of G0 and is greatly affected by variation in Csurf . A
monotonic decrease in the diameter is obtained with lower values of Csurf , rapidly dropping to the lowest value of 0.45 Pa for
the fluid sample with Csurf = 18 mM. We use an inset to Fig. 3(a)
in order to more clearly illustrate the semi-circular shape for
this sample. The fit results (T = 25 ◦ C) are given in Table 1 and
shown in Fig. 4.
In the absence of any solvent contribution to the stress, the
zero shear viscosities of the samples are calculated by
η0 = ηp = G0 λ.
(8)
The dashed lines in Fig. 4(a and b) are best fits from regression analysis and are shown here to qualitatively illustrate the
general trend of the fitted rheological parameters with increasing Csalt . In both cases, increasing salt concentration results in
77
Fig. 3. Cole–Cole plots of EHAC solutions (a) for various surfactant concentrations (Csurf ) and (b) for various salt–surfactant concentration ratios (C* ). The
inset to (a) is the plot for Csurf = 18 mM. The storage modulus, G (ω), and the
loss modulus, G (ω), are determined in a frequency range of 0.01–100 rad/s
with a constant shear strain of 10% (experiments were performed at T = 25 ◦ C).
a monotonic increase in G0 . The variation of plateau modulus
exhibits a power law type behavior, which is expected from theoretical predictions for polymers and micellar solutions [23,69].
The corresponding relations are, respectively, G0 ∼ (Csalt )2.49
for C* = 10.6 held constant (Fig. 4(a)) and G0 ∼ (Csalt )0.73 for
Csurf = 54 mM held constant (Fig. 4(b)). Because the ratio of
surfactant to salt (or equivalently the salinity conditions) are
Table 1
Linear viscoelastic properties of the wormlike micellar fluid samples determined from small-amplitude oscillatory shear experiments T = 25 ◦ C (C* = Csalt /Csurf )
Csurf = 54 mmol/l
E1
C*
Csalt (mmol/l)
λ (s)
G0 (Pa)
η0 (Pa s)
Gmin (Pa)
λbr × 102 (s)
ζ × 104
2.65
143
60.6
4.7
284
0.474
11.2
0.034
5.3
286
171
2.07
355
0.096
112.4
0.428
8.0
432
103
8.7
903
0.246
50.2
0.233
10.6
572
28.4
8.4
238
0.48
19.9
0.494
E2
13.3
718
10.9
11.6
126
0.599
12.6
1.344
E3
16
864
5
11.8
59
0.681
7.9
2.567
C* = 10.6
E1
Csurf (mmol/l)
Csalt (mmol/l)
λ (s)
G0 (Pa)
η0 (Pa s)
Gmin (Pa)
λbr × 102 (s)
ζ × 104
18
191
115
0.45
51.9
0.055
158
1.886
36
382
48.7
3.6
175
0.166
31.7
0.424
54
572
28.4
8.4
238
0.48
19.9
0.494
72
763
23.7
12.6
298
0.34
15.9
0.450
90
954
2.9
30.2
87.6
1.37
3.9
1.951
78
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
Fig. 4. Linear viscoelastic (LVE) parameters and time scales of EHAC solutions: (a) LVE parameters for various Csurf , (b) LVE parameters for various C* , (c) time
scales for various Csurf and (d) time scales for various C* (experiments were performed at T = 25 ◦ C).
kept constant in Fig. 4(a) for a wormlike micellar solution with
C* = 10.6, the data actually corresponds to an increasing surfactant concentration. The observed scaling of G0 ∼ (Csalt )2.49
indicates that the rheological behavior of these EHAC surfactant
systems in this concentration regime is in good agreement with
the theoretically predicted power law exponent of 2.3 obtained
from scaling theory for micellar solutions [69]. Similar scaling
has also been observed for other micellar systems [71]. Raghavan and Kaler [11] have performed similar analyses for their
EHAC-based solutions. Their measurements were made for a
constant molar concentration ratio of C* = 0.5 and for a constant surfactant concentration of Csurf = 60 mM at a temperature
of 60 ◦ C. Their reported values of the power–law exponents for
these two cases (respectively, 2.27 and ∼1) appear to be close
to the ones obtained in the present study.
Fig. 4(b) shows the variation of the linear viscoelastic properties with salt concentration at a constant surfactant concentration. In this case, a ratio of G0 ∼ (Csalt )0.77 is observed, suggesting a rather salinity-insensitive dependence of the plateau
modulus. This is in accordance with the suggestion that an
increasing salt concentration does not lead to strong changes
in the structural composition of the wormlike micelles at this
salinity level; as a consequence, the modulus varies only weakly.
However, as Safran et al. [72] have noted, the screening effect
of the salt counterions with rising concentration may lead to an
alteration of the scission energy of the micelles, and therefore
significant variation in the relaxation time as shown in Fig. 4(d).
At high salinities, Candau et al. [73] describe the formation
of a dynamic three-dimensional network of wormlike micelles
that results in a local maximum in the relaxation time and also in
the resulting viscosity of the solution (see Eq. (8)). We observe
a similar behavior in Fig. 4(d) for the relaxation time and in
Fig. 4(b) for the viscosity. This pronounced maximum is again
in agreement with the viscosity measurements of Raghavan and
Kaler for an EHAC/sodium salicylate solution [11]. However,
it should be noted that with the smaller Cl− ions utilized in
our study, the critical concentrations required for the maxima
in the viscosity and relaxation time to be observed are nearly
a decade higher. A possible explanation for this shift is that, in
contrast to salicylate, the nonbinding character of the Cl− ions
limits the penetration between the head groups of the EHAC
molecules that comprise the micelles, so that this screening
effect requires much higher salt concentrations. The critical concentrations of the samples that correspond to maximum values
of η0 are Csalt = 762 mM (or Csurf = 72 mM) for C* = 10.6 and
Csalt = 332 mM (or C* = 8) for Csurf = 54 mM.
We notice in Fig. 4(b) that for C* ≥ 8 the relationship between
Csalt and η0 becomes almost linear on this log–log plot with a
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
power law exponent, η0 ∼ (Csalt )−3.37 . This value of a power
law exponent is slightly lower than the power–law exponent
of −5 quoted by Larson [1] and Shikata and Kotaka [74] for
some gel-like surfactant systems. Clausen et al. [75] describe
electron microscopy imaging studies of micelles in this region
of decreasing relaxation time that demonstrate that the length
of the micelles (and thus the modulus) remains fairly constant,
indicating that it is indeed the screening by the increased salt concentration that accelerates the micellar reformation processes.
For clarity in our discussion of the nonlinear rheology in the
following section we will focus on the three fluid samples (denoting them by E1, E2 and E3) on the power law curve of Fig. 4(b)
to examine the effects of salt–surfactant ratio on nonlinear shear
and extensional flow behavior. The composition of fluids E1, E2
and E3 along with other micellar fluid samples used in this work
are given in Table 1.
The representation of the storage and loss moduli for the
investigated solutions in the form of Cole–Cole plots (Fig. 3)
shows that in all cases Maxwell-like behavior is obtained in the
low-and medium-frequency regimes. However, in Fig. 3(b), significant deviations from Maxwell behavior are present at high
frequencies. These deviations are expected at high frequencies
because Rouse-like behavior of the individual entangled segments and an additional solvent stress are present. Consequently,
there is an upturn of G as a function of G as noted by Fischer
and Rehage [27]. The addition of a Newtonian ‘solvent-like’
contribution to Eq. (7) with ηs G0 λ, allows a better agreement at high frequency regimes while the low-frequency regime
remains unaffected:
G (ω) = ηs ω + G0
ωλ
.
1 + ω 2 λ2
(9)
A quantitative determination of λbr is possible by fitting
simulations [21] or numerical calculations [20], including the
ratio ζ (from Eq. (3)) as a fitting parameter, to the experimental
Cole–Cole plots. Combining Eq. (3) with Eq. (4) results in the
expression
√
λbr = λ ζ
(10)
and enables the calculation of λbr from the relaxation time and
the ratio ζ. A more accessible method to extract λbr from the
measured deviation of G from the single-mode Maxwell model
was introduced by Kern and co-workers [21,22,76]. According
to the method and Eq. (9), a departure from the semi-circular
shape at high frequencies must be represented with a ‘dip’ and
a ‘tail’ indicating Rouse-like behavior. The dip corresponds to
a local minimum in the loss modulus (Gmin ) and the ratio of
Gmin /G0 is directly correlated to the entanglement length and
average contour length of the micelle. Thus, the critical frequency ω* at which the minimum value of the loss modulus is
obtained provides a much better measure for micellar breakup
time [27]. This approach was used in this report to extract the
micellar breakup timescale λbr from the linear viscoelastic data.
The results, as well as the value of the parameter ζ calculated
from Eq. (10), are listed in Table 1. As mentioned before, the
zero shear rate viscosities of the fluids E1, E2 and E3 change
79
in a power law fashion with salt concentration (see the log–log
plot shown in Fig. 4(b)).
The values of λbr and ζ further support the expected monoexponential behavior since ζ 1, and therefore the approximation
λbr λrep holds for almost all of the fluid samples studied.
The departures from semi-circular response on a Cole–Cole plot
occur at progressively higher frequencies with decreasing values
of ζ as can be seen in Fig. 3. The evolutions of λ and λbr with
increasing salinity are quite similar as shown in Fig. 4(c and
d). Both time constants monotonically decrease with increasing
surfactant concentrations (Csurf ) for a constant salt–surfactant
concentration ratio of C* = 10.6. However, the effect of salt concentration on λ and λbr at a constant surfactant concentration is
more complex. Both the relaxation and micellar breakup times
exhibit a maximum at Csalt = 286 mM (or C* = 5.2) and then
rapidly decrease at higher salinity. On the other hand, the reptation time λrep remains initially unaffected by salt-variation,
but then sharply decreases with increasing salt concentration.
This complex dependence appears to be associated with shearinduced morphology change or phase separation of micellar fluid
samples, as previously observed by Raghavan and co-workers
[12,13].
3.2. Steady and transient shear flows of micellar solutions
3.2.1. Steady shear flow
The nonlinear shear rheology of the micellar solution denoted
E1 (see Table 1) was determined under controlled shear stress
conditions, the results are given in Fig. 5 in the form of a flow
curve and the corresponding viscosity-shear rate curve. Dimensionless shear stress, shear rate and viscosity are, respectively,
defined as τ * = τ/G0 , γ ∗ = γ̇λ, and η* = η/η0 . The symbols correspond to discrete stresses in a continuous stepped stress ramp
experiment (no rest between steps). The stress-increment of each
step is equal in log-space (10 equally spaced points per decade).
All experimental points displayed here were measured for an
overall time of 300 s and represent the averaged signal. The top
branch (hollow circles) corresponds to increasing increments
in τ * and the lower branch (hollow diamonds) to decreasing
increments. Both figures show a distinct hysteresis between the
increasing and decreasing stress ramp. At low stresses the fluid
shows a nearly Newtonian flow behavior. However, above a critical stress τ * ∼ O(1) for an increasing stress ramp the shear rate
increases rapidly and leads to a second Newtonian regime at high
stresses. The huge drop (5000-fold) in the viscosity at this stress
level is shown in Fig. 5(a) and is commonly associated with a
yielding process, or microstructural breakdown [77]. However,
the nature of this structural transition cannot simply be derived
from a single steady stress experiment. As already pointed out
by Cates et al. [40] it is expected for a wormlike micellar system
that an increasing stress ramp experiment will show a monotone increasing shear rate, albeit with a rapid increase of the
∗ . However, the simple reptation theory
shear rate at τ ∗ > τmax
suggests that there will be a maximum of the stress-shear rate
curve as pointed out in the discussion of Fig. 1 and expected
from the constitutive equation of Eq. (1). For an up-ramp stress
experiment the critical stress τmax for the onset of the plateau is
80
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
Fig. 5. The results of up and down ramp controlled stress experiments for the E1
fluid: (a) dimensionless viscosity (η* = η/η0 ) as function of the dimensionless
shear rate (γ ∗ = γ̇λ) and (b) dimensionless stress (τ * = τ/G0 ) as a function of the
dimensionless shear rate. The top branch (hollow circles) corresponds to increasing increments in τ * and the lower branch (hollow diamonds) to decreasing
increments. Solid lines represent results of the single-mode Giesekus constitutive equation (experiments were performed at T = 25 ◦ C).
determined by the top jump condition
τmax ∼
= 0.67G0
(11)
as deduced from reptation theory [24]. A marginally smaller
value of τmax ∼
= 0.54G0 is predicted for nonuniform flow fields
∗ of a down ramp experiment should
[78]. The critical stress τmin
follow a bottom jump condition and result in a hysteresis
between the up and down ramp stress plateau as theoretically
proposed by Porte et al. [49].
The resulting nonmonotonic stress versus shear rate curve
leads for a certain range of stresses τ1∗ < τ ∗ < τ2∗ to a bifurcation
in the flow and the occurrence of nonhomogeneous behavior
in the form of shear-bands. This formation of local spatially
inhomogeneous shear-bands has been experimentally observed
by NMR imaging [79,80], neutron scattering [81,82] and other
flow imaging diagnostics [38,83–89].
The occurrence of an underlying nonmonotonic constitutive
relationship between the steady shear stress and shear rate can
also be directly inferred from the ‘up’ and ‘down’ ramp experiments shown in Fig. 5. The microstructural compositions of the
shear-banded states are highly degenerate and at a stress level of
∗ individual regions may be associated with any deformation
τmax
∗ ≤ γ ∗ ≤ γ ∗ indicated in Fig. 1. Theories which incorporate γM
2
rate a coupling between stress and microstructure are required
in order to provide a selection mechanism for these bands [31].
An increase in the externally imposed driving stress and/or
the time of imposed shearing leads to progressive destruction of
a larger fraction of the wormy micellar (equilibrium) state and
formation of an increasing percentage of the shear-distrupted
micellar segments. These segments contribute to the second
low-viscosity mode that provides the upper Newtonian contribution to the stress. The final stationary flow state can also
strongly depend on the flow history of the micellar solution,
as reported by Radulescu et al. [90]. As the stress is decreased
from the second Newtonian flow regime, which is dominated
by the solvent-like contribution to the total stress, the structural
buildup of the wormlike micelles starts at the stress minimum
∗ indicated in Fig. 1.
τmin
The Johnson–Segalman (JS) equation [91] has been commonly used to model the nonmonotonic material instability and
formation of shear-banding. Greco and Ball [92] suggested a
selection criterion for the symmetry breaking of the band distribution, based on the stress distribution in a circular Couette flow.
Lu et al. [93] considered planar shear for a JS model with additional diffusive terms. However, preliminary attempts to fit the
present shear-flow experiments indicated that the JS model over∗ − τ ∗ ) within a reasonable shear
predicted the value of (τmax
min
rate range of γ1∗ ≤ γ ∗ ≤ γ2∗ . A rederivation of the JS model
from microstructural considerations in order to obtain a full set
of equations with higher order derivative terms and including a
conservation equation for the local number density of micelles
coupled to the stress equations may improve the predictions [31].
A simple way to capture this behavior is given by a number of semi-empirical constitutive equations comparable to
the Johnson–Segalman equation, which incorporate the linear Maxwell model under small strains or strain rates and are
expanded to incorporate additional nonlinearities. Examples of
appropriate nonlinear constitutive equations include the PhanThien and Tanner model [94], the Giesekus model [25,26], the
White-Metzner [95] or the Bird-deAiguiar model [96]. These
models may all exhibit nonmonotonic shear stress-rate behavior
depending on the values of the relevant nonlinear parameters
and the background solvent viscosity.
In the following we will use the single-mode Giesekus
constitutive equation (Eq. (2)) to model the hysteresis of the
shear-banding phenomenon since this equation has already been
demonstrated to model the nonlinear properties of wormlike
micellar solution in start-up of steady shear flows quite well
[28]. The dimensionless viscosity (η* = τ * /γ * ) computed from
the single-mode Giesekus equation is given by:
η∗ =
(1 − f )2
+ βs ,
1 + (1 − 2α)f
(12)
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
81
√
1 − (κ1 − 1)/κ2
√
,
1 + (1 − 2α) (κ1 − 1)/κ2
κ1 = 1 + 16α(1 − α)(γ ∗ )2 ,
f =
(13)
(14)
κ2 = 8α(1 − α)(γ ∗ )2 ,
(15)
where α and βs are, respectively, the anisotropy factor and the
dimensionless solvent viscosity that dominates the viscosity of
the solution at high shear rates. The solid lines in Fig. 5(a and b)
represent the predictions of the model with fitting parameters of
α = 0.93 and ηs = 0.1 Pa s (or βs = 4.18 × 10−4 ). The parameter
βs has been chosen as an adjustable parameter since the viscosity
in the high shear rate regime includes not only the Newtonian
solvent but also the additional, undefined contributions of the
surfactant molecules in a nonmicellar state. Small deviations of
the solution viscosity at high shear rates from the expected complex viscosity (Eq. (9)) of the Rouse-like regime have also been
reported by Manero et al. [14]. The linear viscoelastic rheological parameters given for the E1 solution in Table 1 were used
in the calculations as starting values for a best fit of the model
to the experimental data.
The maximum deviation between experimental values of the
linear viscoelastic parameters and the best fit of the model is
only 0.32%, as seen in Table 2 below. The agreement between
the plateau values of the experimental data and the local minima
and maxima of the theoretical curve is very good, especially in
predicting the local minimum of the stress during the down ramp
of the stress sweep.
The effect of salt–surfactant concentration ratio, C* , on the
hysteresis/shear-band structure can be seen for C* = 13.2 and 16
(fluid samples E2 and E3) in Fig. 6(a and b). The qualitative
features of the flow curves for both fluids are similar to the E1
fluid. However, the distance between the upper and lower branch
∗ − τ ∗ ) decreases as C* increases. Indeed, the magnitude
(τmax
min
of the anisotropy factor in the Giesekus model provides a direct
measure of the width of the hysteresis gap; α decreases from
0.93 for the E1 fluid to 0.89 and 0.85 for the E2 and E3 fluids,
respectively (see Table 2). The stabilizing effect of increasing the
salt–surfactant concentration C* on the micellar fluid structure
was also noted by Raghavan and Kaler [11] and was interpreted
as a promotion of the level of micellar branching. The specific
details of the range of salt–surfactant concentration required for
phase-stabilization is expected to vary with the nature of the
counterion (e.g. whether it is binding or nonbinding). However, there is at present no reported microstructural analysis
Table 2
Giesekus model parameters in shear flow (T = 25 ◦ C)
λ (s)
G0 (Pa)
η0 (Pa s)
α
ηs (Pa s)
a
E1
E2
E3
28.4 (0.07)a
8.42 (0.24)
239 (0.32)
0.93
0.10
11.6 (6.6)
12.3 (6.2)
143 (13.2)
0.89
0.085
4.97 (−0.6)
12.6 (6.4)
62.4 (5.7)
0.85
0.025
The numbers in parentheses show deviation (in percentage) from linear rheology experiments.
Fig. 6. The results of up and down ramp controlled stress experiments: (a) for
the E2 fluid (C* = 13.2) and (b) for the E3 fluid (C* = 16). Solid lines represent
the Giesekus model predictions (experiments were performed at T = 25 ◦ C).
and phase behavior data for the current samples to confirm this
hypothesis.
The observable range of the Newtonian regime after the second upturn in the shear stress also becomes smaller with increasing C* due to a fast transition of the micellar solution to a foamy
thixotropic state that shows a strong increase of the shear stress at
a nearly constant rate. The stress sweep experiments cannot proceed further since irreversible structural changes occur at higher
stresses. This stress-jump behavior at the transition to a shear
induced structure [34,44] limits the ability of the single-mode
Giesekus equation to model the macroscopic stress beyond the
shear-banding regime because the solution is no longer a single
homogeneous phase. All of our experiments are truncated at this
SIS transition.
The effects of temperature on the dynamics of shear-banding
were also investigated for the E1 fluid over a temperature range
of 25–70 ◦ C. The required zero shear viscosities for evaluation
of the Giesekus model predictions were obtained from timetemperature superposition. The temperature dependent shift factor aT (T; T0 ) was determined from a series of ten viscosity
82
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
Fig. 7. Influence of the temperature on hysteresis/shear-band structure for the E1 fluid: (a) T = 30 ◦ C, (b) T = 35 ◦ C, (c) T = 55 ◦ C and (d) T = 70 ◦ C. Solid lines
represent the Giesekus model fits at each temperature.
measurements over low shear rates (γ ∗ ≤ γ1∗ ) in a temperature
range of 25 ◦ C ≤ T ≤ 70 ◦ C and aT follows a simple activated
rate process of Arrhenius form
η(γ̇, T ) T0 ρ(T0 )
H 1
1
aT (T ) =
≈ exp
(16)
−
η(γ̇, T0 ) Tρ(T )
R
T
T0
where aT is the temperature shift function and the density ratio
ρ(T0 )/ρ(T) for the aqueous micellar solution is assumed to be
approximately equal to unity over the investigated temperature
range. For the investigated system with T0 = 25 ◦ C, the activation
energy ratio H/R was found to be 11500 K. The thermal variations of the background solvent viscosity ηs (T) and anisotropy
factor α(T) were determined by an independent fitting procedure.
The variation of these parameters with temperature is also found
to be nearly exponential. The experimental and model results are
given in Fig. 7. The hysteretic behavior becomes progressively
weaker at elevated temperatures, and finally disappears at 70 ◦ C.
The anisotropy factor α in the Giesekus model clearly confirms
this trend and decreases from 0.93 at T = 25 ◦ C to 0.50 at 70 ◦ C.
The value α = 0.5 corresponds to the Leonov model and is the
limiting value between a monotone rise of the stress (with the
absence of the hysteresis behavior) and the development of a
shear-banding region [97].
Visual analysis also shows that the fluid sample remains
homogeneous and rheological data can be measured up to 90 ◦ C.
The stabilizing effect of a moderate level of temperature rise
has previously been observed for viscoelastic flow instabilities
[98,99]. There are also computational studies in the context of
elastic instabilities to confirm this observation. For EHAC based
solutions, the effect of temperature on the degree of micellar
branching is shown to be analogous to increases in salt concentration [12], and hence such a stabilization effect with increasing
temperature is to be expected.
3.2.2. Transient shear flow
The temporal characteristics of the nonmonotone stress shear
rate relation and the hysteresis behavior of the viscoelastic surfactant solution in circular Couette flow can be monitored by
following the transient shear rate evolution at various values of
a constant controlled shear stress.
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
83
Fig. 8. Measurements of temporal fluctuations in the shear rate for various values of the dimensionless shear stress for the E1 fluid. Each transient shear rate γ * (t)
at a given τ * is normalized by the corresponding time-averaged nominal value (
γ * (t)): (i) location of the transient experiments on the qualitative stress shear rate
curve and (ii) transient shear rates for various values of (τ * ; γ * (t)): (a) (0.36; 0.95) (b) (0.42; 1.37), (c) (0.48; 81), (d) (0.60; 1382), (e) (0.72; 3318), (f) (0.84; 3719),
(g) (1.2; 4778) and (h) (1.68; 5514) (experiments were performed at T = 25 ◦ C).
84
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
A series of measurements of the evolution in the transient
dimensionless shear rate of γ * (t) is shown (for E1 fluid) in Fig. 8
for various values of the dimensionless shear stress, τ * . The
dimensionless shear rate γ * (t) is normalized by its time-averaged
nominal value (
γ * (t)) to illustrate the relative magnitude of
the fluctuations around unity. For low values of τ * , the instantaneous shear rate is essentially constant in time (t), as shown
in Fig. 8(a and b) with r.m.s. fluctuations of less than 4.2 and
4.1%, respectively. For τ * < 0.48 the shear flow of the micellar
solution is steady. The flow becomes strongly time-dependent as
the dimensionless stress is increased to τ * = 0.48, as depicted in
Fig. 8(c). The shear rate monotonically increases with time over
the time range of the experiment. The experimental timescale is
restricted to about 600 s at each stress to minimize the effect of
evaporation of the sample in a longer time-period. Grand et al.
[78] noted that at the onset of the plateau region the timescale for
equilibration into the final steady state could be up to two orders
of magnitude higher than the linear viscoelastic relaxation time.
This monotonic increase of the shear rate with time continues
even at higher values of τ * (see Fig. 8(d)) in the nearly horizontal plateau regime of the τ * –γ * plot. Eventually, this transient
increase of the shear rate saturates into a strongly time-periodic
flow around an average shear rate as seen in Fig. 8(e and f) at
the upper end of the plateau regime. The amplitude of the shear
rate fluctuations in this time-periodic flow state can reach up to
50% with respect to the nominal value (or up to 14.8% in terms
of r.m.s. fluctuations). At the highest stresses (τ * ≈ 1), on the
upper branch of the flow curve corresponding to Fig. 8(g and h),
the fluctuations decrease again and are less than, respectively,
15 and 10% (or 9 and 6% in r.m.s. fluctuations) of the nominal
signal.
The characteristic frequencies of the shear rate oscillations
can be quantitatively identified by Fourier analysis of the
time series measurements. Power spectral-densities of γ * (t)
are shown in Fig. 9 for selected values of τ * . Fig. 9(a–d),
respectively correspond to power spectral densities of the signals γ * (t)/
γ * (t) given in Fig. 8(e–h). The power spectrum
obtained from FFT analysis are also normalized by a nominal
component of the spectrum for each τ * , in order to provide consistency in comparison, as suggested by Yesilata et al. [100].
For τ * = 0.72, the power spectrum shows a peak at f1 = 0.032 Hz
(Fig. 9(a)), corresponding to a dimensionless frequency value
of f1∗ = λf1 ≈ 1 since the characteristic relaxation time of the
fluid is λ = 28.4 s. The nominal steady rotation rate of the Couette
fixture is ωc = γ̇/Λ = 2.6 rad/s, where Λ = R1 /d is the ratio of
inner rotor radius to the gap [101]. The nominal rotation rate is
thus nearly one order higher than the value of ω1 = 2πf1 = 0.2 in
rad/s. Wheeler et al. [38] performed similar experiments using
a controlled stress device at a nominally constant deformation
∗ ≤ γ ∗ ≤ γ ∗ . In their experiments, strongly perirate between γM
m
odic oscillations were observed above a critical stress with a
frequency that was dependent on the moment of inertia of the
test (Couette) fixture and the characteristics of the instrument
feedback loop. We see no such coupled instrument/fluid oscillations with the present rheometer (probably due to a tighter
feedback control loop). The periodic frequency identified in
Fig. 9(a) corresponds purely to the fluctuating material response
since the dimensionless frequency is nearly equal to unity
(f1∗ ≈ 1).
For stresses above the critical value for onset of time-periodic
flow, the dominant frequency of oscillations increases and multiple peaks are discernable (Fig. 9(b)). The appearance of these
new peaks in the power spectrum can be considered as the
first sign of the onset of a more complex chaotic-like flow. In
the strongly fluctuating regimes shown in Fig. 9(c and d) the
shear rate oscillates with multiple in commensurate frequencies as determined by the FFT analysis. However, the intensity
of the oscillations decreases, as the linear relationship between
Fig. 9. Fast Fourier transforms (FFT) of γ * (t) in the time-periodic and the time-chaotic regions for E1 fluid. The corresponding dimensionless stresses are given as:
(a) τ * = 0.72, (b) τ * = 0.84, (c) τ * = 1.20 and (d) τ * = 1.68. The spectra are normalized with the nominal value of the spectrum for each case.
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
τ * and γ * (t) is re-established and the stress shear rate curve
approaches the second upper linear flow regime.
The time-dependent fluctuations leading to inhomogeneous
and banded structures were noted recently by Lee et al. [39]
in the middle of the gap of a Couette cell using pointwise flowinduced birefringence (FIB) measurements. Recent studies have
shown that transient oscillations also occur in more complex
flows of micellar fluids with shearing and extensional kinematics
[52,53]. It does appear that these fluctuations are not specific to
a single kinematic field but are common to all flows in which
the characteristic deformation rate and micellar time scale are
on the same order.
3.3. Transient extensional flow of micellar solutions
The transient extensional response of three micellar fluids
(E1, E2 and E3) was also investigated with a capillary breakup
extensional rheometer. The experimental device utilizes plates of
radius R0 = 3.0 mm separated by an initial distance L0 = 2.2 mm
so that the initial aspect ratio is Λ0 = L0 /R0 = 0.733. The plates
are separated rapidly (within 50 ms) to a final separation of
L1 = 6.6 mm.
Fig. 10 shows a sequence of images of the test fluids at
T = 25 ◦ C. The first images show that the initial filament configu-
85
rations for all samples are axially nonuniform. For convenience,
time is referenced to the instant when stretching is halted (i.e.
we set t = t1 = 0). The subsequent images show the progressive
elasto-capillary drainage and ultimate breakup of the filament
at a critical time after the cessation of stretching. The temporal
resolution of the breakup event is limited by the framing rate of
the video camera to 30 frames/s.
The filaments of all three samples shown in Fig. 10 neck
and break in a quantitatively different manner. Both the axial
profiles of the filaments and the time evolution in the midfilament
diameter are significantly different for each type of fluid. The
axial profile of the E1 fluid filament is nearly homogeneous, and
the minimum diameter is always close to the midplane (although
gravitational effects cause the actual minimum to be slightly
above the midplane at initial stages). The effect of gravity at
early times becomes more pronounced for the E2 sample, and
especially for the E3 fluid. However, at later stages, the axial
profile of the filament rapidly evolves into an axially uniform
cylindrical filament. The time to breakup for the E1 fluid is
significantly longer than those of E2 and E3.
Transient midfilament diameters measured for the three fluids are shown in Fig. 11(a). Notice that measurements from three
separate experiments for each fluid agree extremely well and we
obtain excellent experimental reproducibility. Measurements of
the filament evolution for the fluids decay approximately exponentially with time as shown by the solid lines. Prior to the
breakup, significant deviation from the initial exponential curve
occurs caused by the finite extensibility of the micellar structures.
Entov and Hinch [67] have shown that the exponential behavior of the midfilament thinning at intermediate times is related
to a balance of the capillary and elastic forces in the thread and
allows for the calculation of a relaxation time λE in the elongational flow,
t
.
(17)
Dmid (t) ∼ exp −
3λE
For Boger fluids, Anna and McKinley [66] confirmed (in
accordance with the theoretical predictions from Entov and
Hinch [67]) that λE ≈ λ, where λ is the longest relaxation time of
the polymer solution determined from linear viscoelastic measurements. However, in these micellar fluids, the characteristic
relaxation times for transient elongation, extracted from the linear regimes in the semi-logarithmic plots in Fig. 11(a) according
to Eq. (17) and shown in Table 3, are substantially lower (by
Table 3
Giesekus model parameters in extensional flow (T = 25 ◦ C)
λE (s)
G0 (Pa)
η0 (Pa s)
ηs (Pa s)
α
a
Fig. 10. Time-sequences of flow images in the CABER experiments for (a) E1
fluid (C* = 10.6), (b) E2 fluid (C* = 13.2) and (c) E3 fluid (C* = 16) (experiments
were performed at T = 25 ◦ C).
a
E1
E2
E3
9.03
7.98 (−5.0)a
215 (−9.71)
0.10
0.02
0.229
3.63
11.5 (−0.52)
125 (−0.71)
0.085
0.031
0.241
1.65
11.9 (1.62)
59.5 (0.97)
0.025
0.003
0.227
The numbers in parentheses show deviation (in percentage) from linear rheology experiments.
86
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
We now use the simple zero-dimensional formulation proposed by Entov and Hinch [67] to predict the filament thinning
behavior of the micellar fluid samples using the single-mode
Giesekus constitutive equation. Ignoring axial curvature effects,
the temporal evolution of the filament mid-diameter Dmid is
given by a balance of the viscous, elastic and capillary forces
(see Anna and McKinley [66]):
dDmid
1
[(τzz − τrr )Dmid − 2σ]
=
dt
6ηs
(18)
where σ denotes the surface tension of the fluid and τ zz and τ rr
are the additional micellar contributions to the axial and radial
tensile stress components of the total stress tensor τ.
For an ideal uniaxial extensional flow, these stress components are evaluated from the Giesekus constitutive equation (Eq.
(2)), which becomes
λE
dτzz
α 2
+ (1 − 2ε̇λE )τzz +
τ = 2G0 λE ε̇
dt
G0 zz
(19)
λE
dτrr
α 2
τ = −G0 λE ε̇
+ (1 + ε̇λE )τrr +
dt
G0 rr
(20)
where the rate of strain is expressed by
ε̇ = −
2 dDmid
.
Dmid dt
(21)
Combining Eq. (18) with Eqs. (19)–(21) gives a coupled set of
ordinary differential equations (ODEs) for Dmid , τ zz and τ rr . The
Eqs. (18)–(20) can be integrated using standard routines for stiff
ODEs under appropriate initial conditions. The corresponding
initial conditions when the stretching is halted are given by [66]:
Fig. 11. (a) Transient midfilament diameter profiles of the E1, E2 and E3 fluids.
Different symbols correspond to separate CABER experiments for each fluid.
Solid lines show best exponential fits of the curves. (b) Comparison of measured
midfilament diameter profiles (symbols) for micellar fluid filaments with predictions from the single-mode Giesekus model (solid lines). The diameter and
time are scaled with D1 and λE (experiments were performed at T = 25 ◦ C).
almost a factor of three) than the relaxation times obtained from
oscillatory shear flow. We are not aware of any other reports of
the characteristic time constant for extensional flow of micellar solutions; however, Rothstein [65] noted that the apparent
nonlinearity of a micellar network is also characterized by very
different values of the relevant rheological parameter in shear
and extension (in this case the finite extensibility parameter).
This factor of three differences in the characteristic relaxation
time must be associated with the different dynamics of micelle
creation and destruction in the elongated and aligned state that
is associated with extensional flow in a thinning filament as
compared to the fully entangled three-dimensional structure that
exists under equilibrium conditions. Thus, the transient extensional behavior of the wormlike micelles studied here appears
to be more complex than that expected for a simple viscoelastic
fluid of Maxwell–Oldroyd type.
Dmid (t1 = 0) = D1 ,
(22)
2σ
2ηs
τzz (t1 = 0) ∼
−a
,
=
D1
λE
(23)
τrr (t1 = 0) = 0.
(24)
The parameter a in Eq. (23) has been introduced by Anna
and McKinley [66] to correctly account for the initial deformation. The results of simulations (the solid lines), using a and α
as adjusting parameters, along with experimental measurements
(the symbols) are depicted in Fig. 11(b). The time and diameter are scaled with λE and D1 , respectively, and the parameters
used in the calculations are listed in Table 3. It can be seen
from the figure that quantitative agreement between experiments
and theoretical predictions can be obtained, provided that the
model parameters are not forced to be the same as the values
obtained in shear flow. The anisotropy factor α for extensional
flows obtained from the fits is more than a decade smaller than
the value for shear experiments (see Table 2). Most interestingly,
the relaxation times λE obtained from best fits to Eq. (17) are a
factor of almost three smaller than the value expected in shear
flow. This suggests that the arguments leading to Eq. (4) for λ
need to be modified for an elongated micellar chain reptating
and breaking/reforming in an extensional flow.
It is also desirable to re-express the midfilament diameter
data in a more intuitive format, i.e. as a transient tensile stress or
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
extensional viscosity. The thinning dynamics of the elastic fluid
filaments result from an interplay between the fluid rheology and
the effects of capillarity. We have demonstrated that the observed
evolution of the midfilament diameter as the fluid thread necks
down and breaks can be well described using a single-mode
Giesekus model and an appropriate force balance on the fluid
filament, provided that the initial deformation is correctly
accounted for. The transient extensional rheology of the fluid is
encoded in this evolution and we can obtain an apparent extensional viscosity that is related directly to the midfilament diameter by using the same approach described by Anna and McKinley
[66]. An appropriate apparent extensional viscosity calculated
from the surfactant and solvent contributions is given by
(τzz − τrr )
+ 3ηs .
(25)
ε̇
This relation combined with Eq. (18) and (21) gives a direct
dependence of the apparent extensional viscosity from the midfilament diameter evolution
σ
ηE = − dD
(26)
ηE =
mid
dt
87
and we thus differentiate the experimental diameter data to
obtain the transient extensional viscosity. The results are presented in Fig. 12 and show the apparent extensional viscosity in
terms of a dimensionless Trouton ratio (ηE /η0 ) as a function of
the total Hencky strain ε
D1
.
(27)
ε = 2 ln
Dmid (t)
The experimental diameter data obtained from Eq. (26) are
shown as symbols and the Giesekus simulations are indicated by
solid lines. In general, the experimental and theoretical values
for the apparent extensional viscosities extracted from elastocapillary thinning agree well for all three samples. The extensional viscosity at time t1 is a factor of approximately three
higher than the expected extensional viscosity of 3η0 from the
Trouton ratio due to the initial stretch that is imposed by the rapid
extension to the final plate separation prior to the surface tension
driven filament thinning. The extensional viscosity asymptotically reaches a steady state value when the strain becomes large.
However, the final approach to breakup falls below the calibrated minimum resolution of 20 ␮m of the CABER device.
The observed Trouton ratios approaching this steady state are on
the order of 100. Rothstein [65] obtained a similar magnitude
increase in the Trouton ratio for a wormlike micellar solution
system (CTAB/NaSal) in a constant rate filament stretching
experiment at constant deformation rate of ε̇ ∼
= 1.55 s−1 .
4. Conclusions
Fig. 12. Comparison of apparent extensional viscosities computed from midfilament diameter profiles of E1 (䊉), E2 () and E3 () fluids, along with
predictions from the single-mode Giesekus model (solid lines): (a) in log-scale
and (b) in linear-scale.
In the present study, we have examined the linear viscoelasticity, nonlinear shear and extensional flow dynamics for the ternary
system of EHAC/NH4 Cl in salt-free water. EHAC based wormlike micellar solutions are relatively new materials in the context
of wormlike micellar structures and the linear viscoelastic properties have recently been investigated [11,12] in the presence
of different counterions (NaSal, NaCl and NaTos). The present
experiments confirm the earlier observations that EHAC-based
solutions are highly viscoelastic and their structure and rheology delicately depend on the chemical structure and the amount
of salt used as counterion and on the surfactant concentration
and temperature. Although the zero shear rate viscosities of
our solutions at room temperature can be two to three orders
of magnitude lower than those studied previously using different salts, similar nonmonotonic trends in the linear viscoelastic
properties are obtained. As shown in Figs. 2 and 3, the solutions are well-described by a single-mode Maxwell response at
low frequencies (ω<1/λ), but some significant deviations from
˜
this behavior at intermediate
and high frequency regimes are
observed for 5.2 ≤ C* ≤ 8. The variations of the zero shear viscosity and relaxation time as a function of salt concentration
are nonmonotonic as indicated in Fig. 4. These nonmonotonic
rheological responses are not observed in all micellar fluids and
have been attributed to the long, unsaturated chains and complex
headgroups of EHAC-based surfactants [11].
Stepped shear flow experiments reveal the hysteretic behavior of these wormlike micellar fluid samples upon increasing
and decreasing shear stress. This hysteresis can be interpreted
B. Yesilata et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 73–90
88
as an indication of a nonmonotonic underlying flow curve and
the onset of shear-banding at intermediate shear rates. At low
and high shear rates outside this shear-banding regime, the flow
curves measured upon increasing and decreasing shear stress
coincide. The nonmonotonic hysteretic behavior can be modeled
with a single-mode Giesekus constitutive equation, provided that
the anisotropy factor α is greater than 0.5. The hysteretic behavior was investigated under a number of different conditions;
increasing values of salt concentration narrow the hysteresis
region as does an increase in the solution temperature.
The transient evolution of the viscoelastic surfactant solution in steady Couette flow under controlled stress conditions
was explored by conducting a series of instantaneous shear rate
measurements over a wide range of shear stresses. The critical
conditions for the onset of time-dependent flow were determined
quantitatively from these measurements and coincide with the
range of stresses for which hysteretic behavior in the flow curve
is observed.
Experimental observations of the extensional flow behavior
of the micellar fluid samples in capillary breakup experiments
show strong extensional thickening of the samples with the
apparent Trouton ratios increasing by up to two orders of magnitude. The characteristic relaxation times of the fluids determined
from extensional flow experiments are consistently lower than
expected from oscillatory shear flow experiments. A comparison
of the extensional experiments with predictions computed using
the single-mode Giesekus model show good quantitative agreement, however, much lower anisotropy factors (α) are required.
It is thus clear that although a simple nonlinear network
model, such as the Giesekus model captures many of the global
features of wormlike micellar rheology it is not (presently) capable of accurately reflecting the changes in micelle structure that
arise when the kinematics are changed. A more detailed microscopic analysis of the network creation and destruction processes
thus appears warranted. We hope to report on this in the future.
Acknowledgements
The authors would like to thank Schlumberger Cambridge
Research for providing the surfactant fluid samples and Dr. José
Bico for his contribution to the experiments. G.H.M. would also
like to thank the National Science Foundation for partial support
of this work under grant DMS-0406590. B.Y. greatly appreciates
the fund provided for this research by the Turkish Scientific
and Technical Research Institute (under TUBİTAK NATO-B
Fellowship Programme).
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